The Fundamental Equilibrium Equation For Gaseous Stars And The Tolman-Oppenheimer-Volkoff Equation -- Derivations And Applications With Emphasis On Optimisational-Variational Methods

Abstract

Stars are essentially gravitationally stabilised thermonuclear reactors in hydrostatic equilibrium. The fundamental differential equation for all Newtonian gaseous stars in equilibrium is \begin{align} \frac{dp(r)}{dr}=-\frac{\mathscr{G}\mathcal{M}(r)ρ(r)}{r^{2}}\nonumber \end{align} where $p(r),ρ(r)$ are the pressure, density at radius $r$ and $\mathcal{M}(r)$ is the mass contained within a shell of radius $r$ given by $\mathcal{M}(r)=\int_{0}^{r}4π\overline{r}^{2} ρ(\overline{r})d\overline{r}$, and $\mathscr{G}$ is Newton's constant. This simple but crucial differential equation for the pressure gradient within any star, underpins much of astrophysical theory and it can derived by various methods:via a simple heuristic argument; via the Euler-Poisson equations for a self-gravitating fluid/gas; via a variational method by taking the 1st variation of the sum of the thermal and gravitational energies of the star; via the 2nd variation of the Massiue thermodynamic functional for a self-gravitating isothermal perfect-gas sphere; from conservation of the virial tensor; as the non-relativistic limit of the Tolman-Oppenheimer-Volkoff equation (TOVE). The TOVE for equilibrium of relativistic stars in general relativity can in turn be derived by various methods: from the energy-momentum conservation constraint on the Einstein equations applied to a spherically symmetric perfect fluid/gas; via a constrained optimization method on the mass and nucleon number; via a maximum entropy variational method for a sphere of self-gravitating perfect fluid/gas or radiation. An overview is given of all derivations with emphasis on variational methods. Many important applications and astrophysical consequences of the Newtonian equilibrium equation are also reviewed.